How the Secant works
Secant is one of the three reciprocal trigonometric functions. It is defined as the reciprocal of cosine: sec(θ) = 1 / cos(θ). Wherever cosine is small, secant is large, and wherever cosine equals zero, secant is undefined. This calculator takes an angle in either degrees or radians, computes cos(θ), and returns 1/cos(θ) along with the angle expressed in both units.
Formula and method
Given an angle θ, the calculator first converts it to radians if you entered degrees (radians = degrees × π / 180), since JavaScript's trigonometric functions expect radians. It then evaluates cos(θ) and, provided that value is not zero, returns sec(θ) = 1 / cos(θ). The same reciprocal relationship holds for any angle: sec(θ) and cos(θ) always have the same sign, and the reciprocal identity cos(θ) = 1 / sec(θ) can be used to work backward from a known secant value.
Special angle values
- sec(0°) = 1 — since cos(0°) = 1, the smallest possible magnitude for secant
- sec(30°) = 2/√3 ≈ 1.1547 — since cos(30°) = √3/2
- sec(45°) = √2 ≈ 1.4142 — since cos(45°) = √2/2
- sec(60°) = 2 — since cos(60°) = 0.5
Checking your result
Two quick sanity checks apply to every secant result. First, |sec(θ)| should never be less than 1 — if a computed value falls between -1 and 1, the cosine or angle was entered incorrectly. Second, sec(θ) should carry the same sign as cos(θ): positive for angles in the first and fourth quadrants (0° to 90° and 270° to 360°), negative in the second and third (90° to 270°).
Applications
Secant appears throughout trigonometric identities (such as the Pythagorean identity sec²θ = 1 + tan²θ), in solving trigonometric equations, and in fields like optics, surveying, and structural engineering where reciprocal ratios of angles arise naturally — for example, the secant method in numerical analysis and slant-distance calculations both lean on this reciprocal relationship. Label any result with the angle and unit used so it can be reproduced later.